Influence of temperature gradient of slab track on the dynamic responses of the train-CRTS III slab track on subgrade nonlinear coupled system

Temperature is an important load for ballastless track. However, there is little research on the system dynamic responses when a train travels on a ballastless track under the temperature gradient of ballastless track. Considering the moving train, temperature gradient of slab track, gravity of slab track, and the contact nonlinearity between interfaces of slab track, a dynamic model for a high-speed train runs along the CRTS III slab track on subgrade is developed by a nonlinear coupled way in ANSYS. The system dynamic responses under the temperature gradient of slab track with different amplitudes are theoretically investigated with the model. The results show that: (1) The proportions of the initial force and stress caused by the temperature gradient of slab track are different for different calculation items. The initial fastener tension force and positive slab bending stress have large proportions exceeding 50%. (2) The maximum dynamic responses for slab track are not uniform along the track. The maximum slab bending stress, slab acceleration, concrete base acceleration appear in the slab middle, at the slab end, and at the concrete base end, respectively. (3) The maximum accelerations of track components appear when the fifth or sixth wheel passes the measuring point, and at least two cars should be used. (4) The temperature gradient of slab track has a small influence on the car body acceleration. However, the influences on the slab acceleration, concrete base acceleration, fastener tension force are large, and the influence on the slab bending stress is huge.


Coupled dynamic model
The calculation results of the train and track calculated by a 3D train-track-subgrade nonlinear coupled dynamic model with the track and subgrade modeled by 3D solid elements are more precise than those calculated by a 2D train-track on subgrade nonlinear coupled dynamic model using the spring to model the elastic support layer of track to reflect the elastic deformation of the subgrade. However, considering the small time step, large load steps, small mesh size, huge degrees of freedom (DOFs), and nonlinear contact are needed in the simulation, the calculations will be extremely time-consuming. The detailed explanations can be found in Ref. 37 . Moreover, Nguyen et al. 38 compared the dynamic responses of a simplified 2D and full 3D high-speed vehicle-ballasted track on subgrade coupled system, respectively. It concluded that the 2D model using the spring to model the subgrade can meet the precise requirement of the practical engineering and can be used to predict the train and track dynamic responses. Zhai and Cai 39 verified the 2D train-track on subgrade nonlinear coupled dynamic model using the spring to model the elastic support layer of track with the field experiments, the calculated results are close to the measured results. Thus, a simplified 2D train-CRTS III slab track on subgrade coupled dynamic model is adopted in this study. Figure 2 shows the schematic of the coupled dynamic model developed using ANSYS parametric design language. In the model, a high-speed train moves forward along the CRTS III slab track on subgrade at a constant speed V.
The model includes three parts. The wheel-rail interaction submodel based on Hertzian nonlinear contact theory is the same as those in Refs. [39][40][41] . The details of the other two submodels are as follows.
High-speed train submodel. A typical high-speed train has 4 motor and 4 trailer cars. However, according to the research in Ref. 33 , the dynamic responses with 2 cars are almost identical to those with 8 cars. Thus, 2 cars are used in the submodel to improve the computation efficiency of the coupled system.
In Fig. 2, the primary and secondary suspensions are used to connect the bogie and wheel, and the car body and bogie, respectively. The suspensions are simulated by the spring-damper elements. The wheel can only move in vertical motion, while the bogie and car body have two DOFs: pitch and vertical motions. For each car, the DOFs are 10, and the total DOFs for a train with 2 cars are 20. The detailed dynamic equations for a car could be consulted in Refs. 39,40 . The dynamic parameters of the train are given in Table 1.
At the beginning time, the first wheel of the train is located 36.855 m behind the middle of the model. The train moves forward 106.785 m at 300 km/h. The simulation time step is 0.0001 s to reasonably consider the high-frequency dynamic responses.
CRTS III slab track on subgrade submodel. The submodel uses multi-scale modeling technology, as shown in Fig. 2. It consists of three parts. Only fastener and rail are used in the two side parts. While a refined model with a small mesh size is used in the middle part. The lengths of each side and middle parts are 255.465 and 136.08 m, respectively. The total model length is 647.01 m.
The enlarged views of the submodel around the connection between the middle and left side parts, as well as in the middle of the middle part are shown in Fig. 3a and Figure 2. The schematic of the coupled dynamic model.  17.01 m length are modeled, respectively. Small mesh whose length is the 1/6 fastener spacing is adopted to reasonably reflect the complex contact relation between different track components.
Beam elements are employed to model the concrete base, slab, and rail. Spring-damper elements are employed to model the fastener. Contact elements that consider only the compressive force are employed to model the interface between slab and concrete base and the elastic support of the subgrade to reflect the complex time-varying dynamic interlayer contact relation under the combined loads. The submodel parameters are given in Table 2.
Solution and post-processor of the coupled dynamic model. The initial states greatly influence the dynamic characteristics of the coupled system. Therefore, a static analysis is firstly performed under the joint action of train gravity, track temperature gradient, and track gravity. Then, the dynamic simulation for a train moving forward will be performed, taking the static analysis results as the initial conditions. More details on the solution process can be consulted in Refs. 42,43 .
The total node and element numbers in the coupled dynamic model are 5759 and 8291, respectively. The simulation steps for each load case are 11,604. Therefore, it is impossible to store all time histories in the result file due to the limited hard disk volume in an ordinary personal computer. It is vital to choose reasonable postprocess output parameters to obtain a balance between the result file size and the accuracy of maximum dynamic responses. In this study, the nodes and elements in the middle of two adjacent fasteners and at the fastener position from the 9th to 15th slab are chosen as the output range.

Numerical simulation
The dynamic responses of the coupled system have close relation with their initial states. In this Section, the initial states are studied first. Then, considering the influence of the initial states, the envelope curves of slab track as well as the dynamic responses of the coupled system for typical load cases are studied. As can be observed in Figs. 4a-b and 5a-b that the distributions of rail displacement and rotation angle are different. The maximum rail displacement appears at the end or in the middle of slab. However, the maximum rail rotation angle appears near the end of slab. The initial deformations of rail in Fig. 5a-b when f = 90 °C/m are much larger than those in Fig. 4a-b when f = − 45 °C/m. For example, the maximum initial displacement of rail  www.nature.com/scientificreports/ when f = 90 °C/m is 1.058 mm, which is about 3.3 times larger than that when f = − 45 °C/m. The deformed rail when f = 90 °C/m will have a more unfavourable effect on the safety and comfort of the running train. As shown in Fig. 4c, the maximum initial positive and negative rail bending moments appear respectively in the middle and at the end of slab when f = − 45 °C/m. Contrarily, they appear at the end and in the middle of slab, respectively when f = 90 °C/m, as shown in Fig. 5c. It can also be concluded from Figs. 4c and 5c that the initial rail bending moment has some relation with the temperature gradient of slab track. However, the rail stress calculated with the maximum initial rail bending moment 5.161 kN.m is about 15.2 MPa, which is far less than the allowable rail stress 350 MPa 45 . Figures 4d and 5d show the initial fastener forces due to the temperature gradient of slab track. One can find that the maximum initial fastener forces appear at or near the slab end. The initial fastener tension force in Fig. 5d when f = 90 °C/m is 7.096 kN, which is about 3 times larger than that in Fig. 4d when f = − 45 °C/m. The maximum initial fastener tension force when f = 90 °C/m is about 40% of the allowable tension force 18 kN for WJ-8 fastener system 46 and should be taken into consideration in practical engineering.
As shown in Figs. 4e and 5e, the maximum initial negative and positive slab bending stresses occur in the middle of slab. It can also be concluded from Figs. 4e and 5e that the initial slab bending stress in Fig. 5e when f = 90 °C/m is much larger than that in Fig. 4e when f = − 45 °C/m. And the maximum initial slab bending stress when f = 90 °C/m is 2.407 MPa, which is about 85% and close to the allowable concrete tension stress 2.85 MPa for C60 grade concrete in the design code 47 . The slab stress due to the temperature gradient load is large and important for the slab track design and should be considered seriously.
It can be found from Fig. 4f when f = − 45 °C/m and Fig. 5f when f = 90 °C/m that the maximum initial concrete base bending stresses occur in the middle of concrete base and at the slab end in the middle of concrete base, respectively. As can be observed in Figs. 4f and 5f, the maximum initial concrete base bending stresses are 0.253 and 0.564 MPa, respectively, and the stresses are far less than the stresses in Fig. 4e and 5e for the slab.  The influences of temperature gradient of slab track on the maximum initial values for different items are plotted in Fig. 6a-n. It can be concluded from Fig. 6a-n that with the increasing temperature gradient of slab track, most of the maximum initial values for different items will increase. However, the increasing laws for different items are different. The maximum initial upward displacement and rotation angle of rail in Fig. 6a, c, positive and negative rail bending moments in Fig. 6d, e, pressure and tension fastener forces in Fig. 6f, g, gap height under slab in Fig. 6n increase faster and faster. While the maximum initial positive and negative bending stresses of slab and concrete base in Fig. 6h-k, pressure stress under concrete base in Fig. 6m increase slower and slower.
From the results and discussions above, the temperature gradient of slab track affects the initial slab bending stress, fastener force and pressure stress under concrete base significantly. These items will be further analyzed in "Envelope curves of the dynamic responses for typical load cases", "Acceleration time histories and frequency distributions for typical load cases", "Influence of temperature gradient of slab track" sections. Moreover, the high-frequency accelerations of track components due to the slab track's temperature gradient will propagate to the surrounding soil and building and cause environmental vibration. Thus, the accelerations of track components will have a negative effect on the environmental vibration and have acquired more and more attention in the academic and engineering circles and will also be analyzed in "Envelope curves of the dynamic responses for typical load cases", "Acceleration time histories and frequency distributions for typical load cases", "Influence of temperature gradient of slab track" sections.
Envelope curves of the dynamic responses for typical load cases. The envelope curves of the dynamic responses for different calculation items can be obtained by calculating the maximum and minimum It can be concluded that the temperature gradient load greatly affects the dynamic properties of CRTS III slab track. The conclusion is not consistent with that in Ref. 31 . Two reasons may be attributed to this. On the one hand, the contact nonlinearity of structural interface is not considered in Ref. 31 while it is considered in this paper. On the other hand, the ballastless track in Ref. 31 is the CRTS II slab track, which is continuous in longitudinal direction and the temperature deformation is much smaller than the CRTS III slab track in this paper. As shown in Figs. 9a and 10a, the acceleration time history of car body is in regular shape, and the corresponding frequencies for the 4 largest peak points in Figs. 9b and 10b are 14.7, 29.3, 43.9, 58.6 Hz, respectively.   Figs. 4a and 5a. The excitation frequency could be well reflected in the main frequencies of car body acceleration, which could verify the simulation results to some extent. From Figs. 9c, e, g and 10c, e, g, one can find many peaks in the acceleration time history curves, and the peaks appear in the curves when the wheels pass the measuring point. One can further find that the responses are largest when the fifth or sixth wheel passes the measuring point, indicating a large deviation with only one car with 4 wheels considered in the train model. The conclusion is consistent with that in Ref. 33 .

Acceleration time histories and frequency distributions for typical load cases.
From Figs. 9h and 10f, h, it is evident that many high frequencies above 100 Hz appear in the vibration frequency of slab and concrete base. The reason is that there are gaps under the slab, as shown in Figs. 4i and 5i. When a train passes the gap area, due to the dynamic clapping action of slab track, the vibration is large and the vibration frequency is high. The high-frequency vibration is closely related to the gap, so the nonlinear contact element and time step with a small value are needed in the dynamic simulation to truly reflect the dynamic impact effect in the gap area.
The time histories of the wheel-rail force, fastener forces, slab bending stresses, pressure stresses on subgrade when f = − 45 °C/m and f = 90 °C/m are shown in Figs. 11a-f and 12a-f respectively. It should be noted that the time histories in Figs. 11b-f and 12b-f are for the calculation items with the largest responses.
As can be observed in Figs. 11a and 12a, the temperature gradient of slab track has little influence on the wheel-rail force for load case when f = − 45 °C/m. However, its influence on the wheel-rail force for load case when f = 90 °C/m is significant. The reasons can be attributed to that the initial states of slab track when f = − 45   Fig. 4a, i and f = 90 °C/m in Fig. 5a, i are different. In Fig. 5a, i, the initial rail displacement and the gap height under slab are much larger than those in Fig. 4a, i.
From Fig. 11b-f and 12b-f, it can be seen that many peaks appear in the time history curves. The peaks in the front and rear part of the curve can be attributed to the action of the first and second cars, respectively. For example, there are 8 peaks in Fig. 12b, and the corresponding time of the first peak indicates that the first wheel of the train passes the measuring point. The response of the fifth peak in Fig. 12b is the largest, indicating that the maximum response occurs when the fifth wheel of the train passes the measuring point. Many peaks with maximum response appear in the rear part of the curve, indicating that there is some deviation with only one car with 4 wheels considered in the train model.
It is apparent from Figs. 11b-f and 12b-f that initial force and stress exist in the time history curves. However, the proportions of initial force and stress in the total responses differ for different calculation items. The initial fastener tension force and bending stress of slab have a large proportion, while the initial fastener pressure force has a small proportion. For example, the initial and maximum fastener tension forces in Fig. 12c are 7.205 and 11.055 kN, respectively, and the initial force accounts for 65.2% of the maximum force. The initial and maximum positive slab bending stresses in Fig. 12d are 2.405 and 4.605 MPa respectively, and the initial stress accounts for 52.2% of the maximum stress. The initial forces and stresses caused by the temperature gradient and gravity load of slab track can significantly influence the dynamic forces and stresses of slab track and cannot be ignored. The coupled effect of the train, temperature gradient, and gravity of slab track should be considered to obtain reasonable dynamic results of slab track structure.
Influence of temperature gradient of slab track. The influences of temperature gradient of slab track on the maximum dynamic responses for different items of the coupled system are plotted in Fig. 13a-j.
As shown in Fig. 13, the maximum dynamic responses will increase with the increase of the temperature gradient of slab track. However, the increasing laws for different items are different.
As shown in Fig. 13a, the maximum car body acceleration has some relation with the temperature gradient of slab track. However, the maximum car body acceleration 0.046 m/s 2 in Fig. 13a due to the temperature gradient of slab track is only 3.6% of the allowable car body acceleration 0.13 g 48 and can be neglected in practical engineering.  www.nature.com/scientificreports/ As displayed in Fig. 13b-d, the maximum rail, slab, and concrete base accelerations are small when the absolute value of f is less than 45 °C/m. They will increase significantly when f is larger than 45 °C/m due to the influence of the gap under the slab (see Fig. 5i). As illustrated in Fig. 13b-d, the maximum slab acceleration is larger than the rail and concrete base accelerations when f is larger than 45 °C/m. The reason is that the fastener has the vibration-reduction capacity and can reduce the vibration induced by the gap under slab, so the rail acceleration is less than the slab acceleration. The mass of a track slab is much smaller than that of a concrete base, so the concrete base acceleration is also less than the slab. The large vibration of slab track due to large positive temperature gradient load can harm the surrounding environment and should be considered in practical engineering.
It can be observed in Fig. 13e that the temperature gradient of slab track has little influence on the maximum wheel-rail force when the absolute value of f is less than 45 °C/m. And the maximum wheel-rail force will increase rapidly when f is larger than 45 °C/m. Comparing the maximum wheel-rail force when f = 90 °C/m with that when f = 45 °C/m, we can deduce that the increasing rate is 27.3%. The maximum wheel-rail force 89.583 kN in Fig. 13e is far less than the allowable wheel-rail force 170 kN 48 , and the running safety of the train can be ensured.
As shown in Fig. 13f, the temperature gradient of slab track has a small and significant influence on the maximum fastener compressive force when f is smaller than and larger than 45 °C/m respectively. By comparing the maximum fastener compressive force when f = 90 °C/m with that when f = 0 °C/m, it can be calculated that the increasing rate is 52.4%. Comparing Fig. 13g with Fig. 13f, the temperature gradient of slab track has a larger influence on the maximum fastener tension force than the compressive force. By comparing the maximum fastener tension force when f = 90 °C/m with that when f = 0 °C/m, it can be calculated that the increasing rate is 427.4%. The maximum fastener tension force in Fig. 13g is 11.054 kN and is about 61.4% of the allowable fastener tension force 18 kN for WJ-8 fastener system 46 , and the fastener may be damaged under the long-term fatigue load.
As apparent in Fig. 13h-i, the temperature gradient of slab track has a huge influence on the maximum slab bending stresses. The increasing rates for the maximum positive and negative slab bending stresses are 12.79, 7.50 times, respectively. Further analyses show that the maximum slab bending stress in Fig. 13h exceeds the concrete tension stress 2.85 MPa for C60 grade concrete in the design code 47 , and prestress technology and reinforcing bars should be used to improve the durability of slab.
As apparent in Fig. 13j, the temperature gradient of slab track has a large influence on the maximum pressure stress on subgrade. By comparing the maximum pressure stress on subgrade when f = 90 °C/m with that when f = 0 °C/m, it can be deduced that the increasing rate is 1.33 times. It can also be deduced that the maximum pressure stress on subgrade in Fig. 13j is about 54.4% of the allowable pressure stress on subgrade 49 and should be considered in practical engineering.

Conclusions
In this paper, considering the contact nonlinear, the influences of temperature gradient of slab track on the dynamic characteristics of the coupled system are theoretically studied using a coupled nonlinear dynamic model. The following conclusions are drawn.
(1) The increasing laws of the maximum initial value for different items of CRTS III slab track on subgrade are different. With the increase of the temperature gradient of slab track, the maximum initial upward www.nature.com/scientificreports/ displacement and rotation angle of rail, pressure and tension fastener forces, gap height under slab increase faster and faster. While the maximum initial positive and negative bending stresses of slab and concrete, pressure stress under concrete base increase slower and slower. (2) The proportions of initial force and stress in the total dynamic responses differ for different calculation items. The fastener tension force and positive slab bending stress have large proportions exceeding 50%. Therefore, the coupled effect of the moving train, temperature gradient of slab track, and gravity of slab track should be considered. (3) There will appear gaps in the slab track when the temperature gradient of slab track is large. There are many high frequencies above 100 Hz in the vibration frequency of track components, and the nonlinear contact elements are needed in the simulation model to truly reflect the high-frequency dynamic open and closure clapping action between the track components. (4) The distributions of the maximum track dynamic responses are not uniform along the track. Generally, the most unfavorable position for each track component is either at the end or in the middle. The maximum slab bending stress, slab acceleration, concrete base acceleration appear in the slab middle, at the slab end, and at the concrete base end, respectively. (5) The maximum accelerations of track components appear when the fifth or sixth wheel passes the measuring point. There is a large deviation with only one car with 4 wheels considered in the train model, and at least two cars should be used in the train model. (6) The temperature gradient of slab track has different influence laws on the maximum system dynamic responses for different items. It has a small influence on the maximum car body acceleration. However, the influences on the slab acceleration, concrete base acceleration, fastener tension force are large, and the influence on the slab bending stress is huge.

Data availability
The datasets used and analysed during the current study are available from the corresponding author on reasonable request.